#72209649

order	10752
isoclinism	isoclinic class of #72209649
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting

C₄³ ⋊ GL(3,𝔽₂)

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	G	C₁
Frattini, socle	C₂³	AGL(3,𝔽₂)
Fitting, cosocle	C₄³	GL(3,𝔽₂)
LCS	G
UCS	C₁

422 conjugacy classes of subgroups; only the 4 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 4 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Permutation generators:

(1,2)(3,10)(4,8)(5,9)(7,14)(11,12)(15,21)(16,20)(17,22)(23,24);(2,6,7)(8,11,16)(9,12,15)(10,13,17)(14,18,19)(20,23,26)(21,24,27)(22,25,28);(1,3)(2,8,10,9)(4,5)(7,15,17,16)(18,23,25,24)(19,26,28,27);(1,3)(2,8,10,9)(4,5)(6,11,13,12)(7,17)(14,20,22,21)(15,16)(18,25)(19,27,28,26)(23,24);(1,4,3,5)(6,13)(11,12)(14,20,22,21)(18,24,25,23)(19,27,28,26);(2,10)(7,17)(8,9)(15,16)(18,25)(19,28)(23,24)(26,27);(2,10)(6,13)(8,9)(11,12)(14,22)(19,28)(20,21)(26,27);(1,3)(4,5)(14,22)(18,25)(19,28)(20,21)(23,24)(26,27)

Conjugacy classes & character table

19 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72209804, #72209812, #72209820

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((1,2)(3,10)(4,8)(5,9)(7,14)(11,12)(15,21)(16,20)(17,22)(23,24), (2,6,7)(8,11,16)(9,12,15)(10,13,17)(14,18,19)(20,23,26)(21,24,27)(22,25,28), (1,3)(2,8,10,9)(4,5)(7,15,17,16)(18,23,25,24)(19,26,28,27), (1,3)(2,8,10,9)(4,5)(6,11,13,12)(7,17)(14,20,22,21)(15,16)(18,25)(19,27,28,26)(23,24), (1,4,3,5)(6,13)(11,12)(14,20,22,21)(18,24,25,23)(19,27,28,26), (2,10)(7,17)(8,9)(15,16)(18,25)(19,28)(23,24)(26,27), (2,10)(6,13)(8,9)(11,12)(14,22)(19,28)(20,21)(26,27), (1,3)(4,5)(14,22)(18,25)(19,28)(20,21)(23,24)(26,27));